Bourgain [1]} proved that any \((\delta , \sigma )\) -set cannot be approximately closed under both addition and multiplication. More precisely, if \(A\subset [1, 2]\)  is a \((\delta , \sigma )\) -set, then there exists \(\epsilon =\epsilon (\sigma )>0\)  such that
\(\max \left\lbrace |A+A|, |A\cdot A|\right\rbrace \gtrsim \delta ^{-\epsilon }|A|.\) 
